QuestionCalculate the following operations in the given bases:
a. $23_{\text {five }} \cdot 2$ five, b. $23_{\text {five }} + 2$ five, c. $43_{\text {six }} \cdot 25_{\text {six }}$ , d. $123_{\text {five }} \div 2$ five, e. $10010_{\text {two }} + 10_{\text {two }}$ , f. $10011_{\text {two }} \cdot 110_{\text {two }}$ .

Studdy Solution

STEP 1

Assumptions1. We are performing operations in different bases, not the usual base10. . The operations are multiplication, division, and addition.
3. We need to convert the results back to the original base.

STEP 2

Let's start with the first operation, $23_{\text {five }} \cdot2$ five. In base five,2 is still2. So, we can directly multiply.
$23_{\text {five }} \cdot2_{\text {five }}$

STEP 3

Multiply each digit of the number $23_{\text {five }}$ by2.
$2 \cdot3_{\text {five }} =6_{\text {ten }} =11_{\text {five }}$ $2 \cdot2_{\text {five }} =_{\text {ten }} =_{\text {five }}$

STEP 4

Combine the results to get the final answer.
$23_{\text {five }} \cdot2_{\text {five }} =46_{\text {five }}$

STEP 5

Now let's perform the second operation, $123_{\text {five }} \div2$ five. In base five,2 is still2. So, we can directly divide.
$123_{\text {five }} \div2_{\text {five }}$

STEP 6

Divide each digit of the number $123_{\text {five }}$ by2. Remember to convert the results back to base five.
$1_{\text {five }} \div2_{\text {five }} =0.5_{\text {ten }} =2.5_{\text {five }}$ $2_{\text {five }} \div2_{\text {five }} =1_{\text {ten }} =1_{\text {five }}$ $3_{\text {five }} \div2_{\text {five }} =1.5_{\text {ten }} =3.5_{\text {five }}$

STEP 7

Combine the results to get the final answer.
$123_{\text {five }} \div2_{\text {five }} =61_{\text {five }}$

STEP 8

Now let's perform the third operation, $23_{\text {five }}+2$ five. In base five,2 is still2. So, we can directly add.
$23_{\text {five }}+2_{\text {five }}$

STEP 9

Add the number $23_{\text {five }}$ and2. Remember to convert the results back to base five.
$3_{\text {five }} +2_{\text {five }} =5_{\text {ten }} =_{\text {five }}$ $2_{\text {five }} +_{\text {five }} =2_{\text {ten }} =2_{\text {five }}$

STEP 10

Combine the results to get the final answer.
$23_{\text {five }}+2_{\text {five }} =30_{\text {five }}$

STEP 11

Now let's perform the fourth operation, $10010_{\text {two }}+10_{\text {two }}$ . In base two,10 is equal to in base ten. So, we can directly add.
$10010_{\text {two }}+10_{\text {two }}$

STEP 12

Add the number $10010_{\text {two }}$ and $10_{\text {two }}$ . Remember to convert the results back to base two.
$0_{\text {two }} +0_{\text {two }} =0_{\text {ten }} =0_{\text {two }}$ $_{\text {two }} +_{\text {two }} =2_{\text {ten }} =10_{\text {two }}$ $0_{\text {two }} +0_{\text {two }} =0_{\text {ten }} =0_{\text {two }}$ $0_{\text {two }} +0_{\text {two }} =0_{\text {ten }} =0_{\text {two }}$ $_{\text {two }} +0_{\text {two }} =_{\text {ten }} =_{\text {two }}$

STEP 13

Combine the results to get the final answer.
$10010_{\text {two }}+10_{\text {two }} =10100_{\text {two }}$

STEP 14

Now let's perform the fifth operation, $43_{\text {six }} \cdot25_{\text {six }}$ . In base six,43 is equal to31 in base ten and25 is equal to17 in base ten. So, we can directly multiply.
$43_{\text {six }} \cdot25_{\text {six }}$

STEP 15

Multiply the number $43_{\text {six }}$ and $25_{\text {six }}$ . Remember to convert the results back to base six.
$3_{\text {six }} \cdot5_{\text {six }} =15_{\text {ten }} =23_{\text {six }}$ $4_{\text {six }} \cdot2_{\text {six }} =8_{\text {ten }} =12_{\text {six }}$

STEP 16

Combine the results to get the final answer.
$43_{\text {six }} \cdot25_{\text {six }} =1235_{\text {six }}$

STEP 17

Now let's perform the sixth operation, $10011_{\text {two }} \cdot110_{\text {two }}$ . In base two,10011 is equal to19 in base ten and110 is equal to6 in base ten. So, we can directly multiply.
$10011_{\text {two }} \cdot110_{\text {two }}$

STEP 18

Multiply the number $10011_{\text {two }}$ and $110_{\text {two }}$ . Remember to convert the results back to base two.
$_{\text {two }} \cdot0_{\text {two }} =0_{\text {ten }} =0_{\text {two }}$ $_{\text {two }} \cdot_{\text {two }} =_{\text {ten }} =_{\text {two }}$ $0_{\text {two }} \cdot_{\text {two }} =0_{\text {ten }} =0_{\text {two }}$ $0_{\text {two }} \cdot0_{\text {two }} =0_{\text {ten }} =0_{\text {two }}$ $_{\text {two }} \cdot_{\text {two }} =_{\text {ten }} =_{\text {two }}$

STEP 19

Combine the results to get the final answer.
$10011_{\text {two }} \cdot110_{\text {two }} =11000_{\text {two }}$ The final answers area. $46_{\text {five }}$ d. $61_{\text {five }}$ b. $30_{\text {five }}$ e. $10100_{\text {two }}$ c. $1235_{\text {six }}$ f. $11000_{\text {two }}$

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Studdy Solution

STEP 1

Assumptions1. We are performing operations in different bases, not the usual base10. . The operations are multiplication, division, and addition.
3. We need to convert the results back to the original base.

STEP 2

Let's start with the first operation, $23_{\text {five }} \cdot2$ five. In base five,2 is still2. So, we can directly multiply.
$23_{\text {five }} \cdot2_{\text {five }}$

STEP 3

Multiply each digit of the number $23_{\text {five }}$ by2.
$2 \cdot3_{\text {five }} =6_{\text {ten }} =11_{\text {five }}$ $2 \cdot2_{\text {five }} =_{\text {ten }} =_{\text {five }}$

STEP 4

Combine the results to get the final answer.
$23_{\text {five }} \cdot2_{\text {five }} =46_{\text {five }}$

STEP 5

Now let's perform the second operation, $123_{\text {five }} \div2$ five. In base five,2 is still2. So, we can directly divide.
$123_{\text {five }} \div2_{\text {five }}$

STEP 6

STEP 7

Combine the results to get the final answer.
$123_{\text {five }} \div2_{\text {five }} =61_{\text {five }}$

STEP 8

Now let's perform the third operation, $23_{\text {five }}+2$ five. In base five,2 is still2. So, we can directly add.
$23_{\text {five }}+2_{\text {five }}$

STEP 9

Add the number $23_{\text {five }}$ and2. Remember to convert the results back to base five.
$3_{\text {five }} +2_{\text {five }} =5_{\text {ten }} =_{\text {five }}$ $2_{\text {five }} +_{\text {five }} =2_{\text {ten }} =2_{\text {five }}$

STEP 10

Combine the results to get the final answer.
$23_{\text {five }}+2_{\text {five }} =30_{\text {five }}$

STEP 11

Now let's perform the fourth operation, $10010_{\text {two }}+10_{\text {two }}$ . In base two,10 is equal to in base ten. So, we can directly add.
$10010_{\text {two }}+10_{\text {two }}$

STEP 12

STEP 13

Combine the results to get the final answer.
$10010_{\text {two }}+10_{\text {two }} =10100_{\text {two }}$

STEP 14

Now let's perform the fifth operation, $43_{\text {six }} \cdot25_{\text {six }}$ . In base six,43 is equal to31 in base ten and25 is equal to17 in base ten. So, we can directly multiply.
$43_{\text {six }} \cdot25_{\text {six }}$

STEP 15

Multiply the number $43_{\text {six }}$ and $25_{\text {six }}$ . Remember to convert the results back to base six.
$3_{\text {six }} \cdot5_{\text {six }} =15_{\text {ten }} =23_{\text {six }}$ $4_{\text {six }} \cdot2_{\text {six }} =8_{\text {ten }} =12_{\text {six }}$

STEP 16

Combine the results to get the final answer.
$43_{\text {six }} \cdot25_{\text {six }} =1235_{\text {six }}$

STEP 17

Now let's perform the sixth operation, $10011_{\text {two }} \cdot110_{\text {two }}$ . In base two,10011 is equal to19 in base ten and110 is equal to6 in base ten. So, we can directly multiply.
$10011_{\text {two }} \cdot110_{\text {two }}$

STEP 18

STEP 19

Combine the results to get the final answer.
$10011_{\text {two }} \cdot110_{\text {two }} =11000_{\text {two }}$ The final answers area. $46_{\text {five }}$ d. $61_{\text {five }}$ b. $30_{\text {five }}$ e. $10100_{\text {two }}$ c. $1235_{\text {six }}$ f. $11000_{\text {two }}$

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Studdy Solution

STEP 1

Assumptions1. We are performing operations in different bases, not the usual base10. . The operations are multiplication, division, and addition.3. We need to convert the results back to the original base.

STEP 2

Let's start with the first operation, 23five ⋅223_{\text {five }} \cdot223five ​⋅2 five. In base five,2 is still2. So, we can directly multiply.23five ⋅2five 23_{\text {five }} \cdot2_{\text {five }}23five ​⋅2five ​

STEP 3

STEP 4

Combine the results to get the final answer.23five ⋅2five =46five 23_{\text {five }} \cdot2_{\text {five }} =46_{\text {five }}23five ​⋅2five ​=46five ​

STEP 5

Now let's perform the second operation, 123five ÷2123_{\text {five }} \div2123five ​÷2 five. In base five,2 is still2. So, we can directly divide.123five ÷2five 123_{\text {five }} \div2_{\text {five }}123five ​÷2five ​

STEP 6

STEP 7

Combine the results to get the final answer.123five ÷2five =61five 123_{\text {five }} \div2_{\text {five }} =61_{\text {five }}123five ​÷2five ​=61five ​

STEP 8

Now let's perform the third operation, 23five +223_{\text {five }}+223five ​+2 five. In base five,2 is still2. So, we can directly add.23five +2five 23_{\text {five }}+2_{\text {five }}23five ​+2five ​

STEP 9

STEP 10

Combine the results to get the final answer.23five +2five =30five 23_{\text {five }}+2_{\text {five }} =30_{\text {five }}23five ​+2five ​=30five ​

STEP 11

Now let's perform the fourth operation, 10010two +10two 10010_{\text {two }}+10_{\text {two }}10010two ​+10two ​. In base two,10 is equal to in base ten. So, we can directly add.10010two +10two 10010_{\text {two }}+10_{\text {two }}10010two ​+10two ​

STEP 12

STEP 13

Combine the results to get the final answer.10010two +10two =10100two 10010_{\text {two }}+10_{\text {two }} =10100_{\text {two }}10010two ​+10two ​=10100two ​

STEP 14

Now let's perform the fifth operation, 43six ⋅25six 43_{\text {six }} \cdot25_{\text {six }}43six ​⋅25six ​. In base six,43 is equal to31 in base ten and25 is equal to17 in base ten. So, we can directly multiply.43six ⋅25six 43_{\text {six }} \cdot25_{\text {six }}43six ​⋅25six ​

STEP 15

STEP 16

Combine the results to get the final answer.43six ⋅25six =1235six 43_{\text {six }} \cdot25_{\text {six }} =1235_{\text {six }}43six ​⋅25six ​=1235six ​

STEP 17

STEP 18

STEP 19

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Assumptions1. We are performing operations in different bases, not the usual base10. . The operations are multiplication, division, and addition.
3. We need to convert the results back to the original base.

Let's start with the first operation, $23_{\text {five }} \cdot2$ five. In base five,2 is still2. So, we can directly multiply.
$23_{\text {five }} \cdot2_{\text {five }}$

Combine the results to get the final answer.
$23_{\text {five }} \cdot2_{\text {five }} =46_{\text {five }}$

Now let's perform the second operation, $123_{\text {five }} \div2$ five. In base five,2 is still2. So, we can directly divide.
$123_{\text {five }} \div2_{\text {five }}$

Combine the results to get the final answer.
$123_{\text {five }} \div2_{\text {five }} =61_{\text {five }}$

Now let's perform the third operation, $23_{\text {five }}+2$ five. In base five,2 is still2. So, we can directly add.
$23_{\text {five }}+2_{\text {five }}$

Combine the results to get the final answer.
$23_{\text {five }}+2_{\text {five }} =30_{\text {five }}$

Now let's perform the fourth operation, $10010_{\text {two }}+10_{\text {two }}$ . In base two,10 is equal to in base ten. So, we can directly add.
$10010_{\text {two }}+10_{\text {two }}$

Combine the results to get the final answer.
$10010_{\text {two }}+10_{\text {two }} =10100_{\text {two }}$

Now let's perform the fifth operation, $43_{\text {six }} \cdot25_{\text {six }}$ . In base six,43 is equal to31 in base ten and25 is equal to17 in base ten. So, we can directly multiply.
$43_{\text {six }} \cdot25_{\text {six }}$

Combine the results to get the final answer.
$43_{\text {six }} \cdot25_{\text {six }} =1235_{\text {six }}$